On the other hand, if \(({\mathcal {M}}, T, \Omega )\)  is a Borchers triple, then
one can define a net as follows: in the two-dimensional Minkowski space any double cone \(D\)  is
represented as the intersection of two wedges \(D = (W_{\mathrm {R}}+ a) \cap (W_{\mathrm {L}}+ b)\) . Then we define
for double cones \({\mathcal {A}}(D) := {\hbox{\rm Ad\,}}T(a)({\mathcal {M}}) \cap {\hbox{\rm Ad\,}}T(b)({\mathcal {M}}^{\prime })\) . For a general region \(O\)  we take
\({\mathcal {A}}(O) := \bigvee _{D \subset O} {\mathcal {A}}(D)\) , where the union runs all the double cones contained in
\(O\) . Then one can observe that \({\mathcal {A}}\)  satisfies isotony and locality.
Borchers further proved that the representation \(T\)  of \({\mathbb {R}}^2\)  extends to a representation \(U\)  of
\({{\mathcal {P}}^\uparrow _+}\)  through Tomita-Takesaki theory [1]} such that \({\mathcal {A}}\)  is covariant
and \(\Omega \)  is invariant under \(U\)  (the representation of boosts is given
by the modular group). Positivity of energy is inherited from \(T\)  of the Borchers triple.
The only missing property is cyclicity of \(\Omega \)  for local algebras \({\mathcal {A}}(O)\) .
Indeed, there are examples of Borchers triples which fail to satisfy this cyclicity [2]}.
